A344051 a(n) = Sum_{k=0..n} binomial(n, k)*|Lah(n, k)|. Binomial convolution of the unsigned Lah numbers A271703.

1, 1, 5, 37, 361, 4301, 60001, 954325, 16984577, 333572041, 7151967181, 165971975621, 4139744524345, 110333560295557, 3126749660583641, 93819198847833061, 2969676820062708481, 98843743790129998865, 3449675368718647501717, 125921086600579132143781, 4796519722094585691925961

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = n * n! * hypergeom([1 - n, 1 - n], [2, 2], 1] for n >= 1.

D-finite with recurrence +16*n*a(n) +6*(-8*n^2+5*n-1)*a(n-1) +(48*n^3-266*n^2+407*n-167)*a(n-2) +(-16*n^4+106*n^3-219*n^2+108*n+93)*a(n-3) +(n-4)*(2*n^3-13*n^2+16*n+25)*a(n-4) -(n-5)*(n-4)^3*a(n-5)=0. - R. J. Mathar, Jul 27 2022

a(n) ~ n^(n - 1/2) / (sqrt(6*Pi) * exp(n - 3*n^(2/3) + n^(1/3) - 1/3)) * (1 + 31/(54*n^(1/3))). - Vaclav Kotesovec, Apr 27 2024

Maple

aList := proc(len) local lah;
lah := (n, k) -> `if`(n = k, 1, binomial(n-1, k-1)*n!/k!):
seq(add(binomial(n, k)*lah(n, k), k = 0..n), n = 0..len-1) end:
lprint(aList(22));

Mathematica

a[n_] := n n! HypergeometricPFQ[{1 - n, 1 - n}, {2, 2}, 1]; a[0] := 1;
Table[a[n], {n, 0, 20}]

Crossrefs

Cf. A271703, A111596, A344050.

Keyword

nonn

Author

Peter Luschny, May 10 2021

Status

approved